Based on a new algorithm variant of the conceptual design modeling,
proposed by the authors, an application of this algorithm is analyzed on
a product of motor--planetary reducer type. According to the new
variant, the paper exemplifies the generation of the reducer solving
structural variants and the reducer concept establishment.

This paper exemplifies an application of this algorithm on a
product of motor-reducer type, which consists of a planetary speed
reducer and an electric motor. This motor-reducer is used in a
rotational platform on which a parabolic antenna is installed; the
motor-reducer must drive the platform in different angular positions,
under imposed conditions of accuracy and stability. The first steps of
the algorithm up to the generation of solving structural variants
inclusive are approached in this paper.

2. On the requirements list

The following requirements (main objectives) regarding the
rotational platform and the motor-reducer come out from the requirements
list of the global product formed by the parabolic antenna assembly:

1) The maximum diameter of the crown gear that is fixed to the
platform: [approximately equal to] 5 m.

The following four optimization objectives that are enumerated in
their relative importance order are associated to these requirements:

A. Minimization of the production costs,

B. Reducing the friction losses,

C. Minimization of the radial size and

D. Minimization of the axial size.

Afterwards, these (secondary) objectives are used as technical and
economical evaluation criteria for the identification of the principle
solution (among the solving structures that were generated). It is
considered that the weight of the criteria is in the following
correlations: A [approximately equal to] 4B [approximately equal to] 6C
[approximately equal to] 8D.

3. Identification of the motor-reducer function

The motor-reducer global function is a sub-function of the
rotational platform. In order to identify it, in Fig. 1 the following
elements were successively represented:

* The platform global function (Fig. 1,a),

* Its structure of 1M+1E+1I order (Fig. 1,b) and

* The structure of 3M+4E+4I order (Fig. 1,c and d), derived from
the previous structure.

The last structure, illustrated both in a descriptive variant (Fig.
1,c) and in a VDI symbolic variant (Fig. 1,d), has the following
component sub-functions:

* FM1: connecting the material (platform) to the mechanical energy;

* FM2: the material (platform) angular displacing (rotating);

* FM3: registration of the material (platform) angular position;

* FE1: Connecting/disconnecting the voltage;

* FE2: Transformation of the electrical energy into mechanical
energy and its irreversible transmission, together with the speed
primary reducing;

* FE3: Transmission of mechanical energy with the speed secondary
reducing (using the crown gear that is fixed to the platform);

* FE4: Ramification of the mechanical energy into two branches:
useful energy and lost energy (through friction, mainly);

* FI1: Ramification of the input information into two branches:
emission of the starting signal (after fulfilling the exploitation and
safety instructions) and transmission of the information regarding the
wanted angular position to the FI3 (sub)function;

* FI2: Commanding the execution of the starting and stopping
signals; unlike the logical function FE1 (of AND type), the function FI2
is a logical function of INHIBITION type.

* FI3: Reception of the information regarding the angular
positions: wanted and current and their comparison;

* FI4: Emission of the stopping signal when the two positions
become equal, and registration of the final angular position.

Thus, the motor-reducer's global function is represented by
the sub-function FE2, from the structure of 3M+ 4E+ 4I order (Fig. 1,c
and d).

4. Detailing of the motor-reducer function

First, the motor-reducer function (see Fig. 2,a) from Fig. 1,d is
extracted. The structure of functions in a symbolic variant from Fig.
2,b results by detailing this global function.

The three sub-functions of this structure have the following
meanings:

* FE21: Transformation of the electrical energy into mechanical
energy (a main sub-function);

* FE22: Irreversible transmission of the mechanical energy (a main
sub-function as well);

* FE24: Transmission of the mechanical energy (unchanged) between
the satellite-gear and central shaft (a main sub-function).

Further, based on the structure of functions from Fig. 2,b, the
motor-reducer's solving structures are generated.

5. The generation of the solving structures

This stage contains two parts that are relatively distinct:

* Generation (synthesis) of the solving structural variants, and

* Establishment of the solving structures, by the kinematical
configuration (synthesis) of the obtained variants and by the
elimination of the variants whose technical characteristics don't
fulfill quantitatively the requirements from the list.

5.1 Generation of the solving structural variants

From the research on the sources of existent solutions (catalogues
of physical effects, catalogues of solutions for the functions of large
use, prospects, patents, offers etc.), the following conclusions
regarding the sub-functions FE21, ..., FE24 (from Fig. 2,b) solving
result:

* All these sub-functions are of SPK type;

* The potential principle solutions of these sub-functions (the
principles and the solving structures) are entirely known in the effects
plan and are partially known in the plan of the effects carriers'
configuration (a part of the found configurations can be reconfigured).

For exemplification, the results that were found are systematized
in the simplified morphological matrix from Fig. 2,c. By means of this
matrix, more solving structural variants can be generated by combining
and by a compatible composition of the potential solutions from Fig.
2,c; obviously, from the obtained variants, there will be considered as
solving structures (for the motor-reducer's FE2 function) only
those whose technical characteristics fulfill the requirements from the
list quantitatively.

The qualitative schemes of the six variants are represented in Fig.
3,a, .., f in a simplified way (without representing the motor). For
these schemes there are further established the numbers of teeth (from
the condition [absolute value of i] = 100 [+ or -] 1.5 %) and, then, the
efficiency and the moment amplification ratio is calculated for each of
the variants. Obviously, the solving structures of the function FE2 will
be nominated by the variants that obtain: [absolute value of i] = 100 [+
or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible
transmission of the mechanical energy.

[FIGURE 3 OMITTED]

5.2. Establishment of the solving structures

In order to establish the solving structures (from the illustrated
variants), first the synthesis of the number of teeth is made from the
condition: [absolute value of i] = 100[+ or -]1.5%. Then, on the basis
of the known efficiencies of the gear pairs with fixed axes, the
efficiencies of the proposed reducers are calculated in the two possible
actuation cases (direct and inverse) and the amplification ratio of the
input moment for the direct actuation is established.

If the efficiency for the inverse actuation is null or negative,
then the analyzed reducer transmits the power irreversibly and,
therefore, the motor's brake becomes superfluous.

The case of the SR1 and SR2 variants. For each of the planetary
reducers from Fig. 3,a and b (consisting of an involute internal gear
pair 1-2 and of a homokinematical coupling 2-3), the condition of
obtaining the transmission ratio can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [i.suub.0] denotes the kinematical interior ratio of the
planetary gear with a sun gear from Fig. 3,a and b:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

On the limit (for the internal involute gear pairs), admitting that
[z.sub.1] = [z.sub.1] + 4, the following values (see also Fig. 4) are
obtained from relations (1) and (2): [z.sub.1] = 400, [z.sub.2] = 396
and [i.sub.0] = + 0.99. Considering that each of the planetary gears
from Fig. 3,a and b has the interior efficiency:

* the manufacturing technology is relatively simple, but needs high
accuracies. The first three properties show that the requirements from
the list [absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than
or equal to] 0.4 and the irreversible transmission of power) are
fulfilled without motor's brake; therefore, each of the variants
SR1 and SR2 is a solving structure of the function FE2.

The case of SR3 and SR4 variants. For each of the planetary
reducers from Fig. 3,c and d (consisting of the homo-kinematical
coupling 1-2 and the cycloidal gear pair with rollers 2-3), the
condition of obtaining the transmission ratio can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [i.sub.0] denotes the interior kinematical ratio of the
planetary gear with a sun gear (Fig. 3,c and d):

* the manufacturing technology uses special machine tools with high
accuracies.

The first three properties show that the requirements from the list
([absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal
to] 0.4 and the irreversible transmission of power) are fulfilled;
therefore, each of the variants SR3 and SR4 is also a solving structure
of the function FE2.

The case of the SR5 and SR6 variants. The planetary chain reducers
from Fig. 3,e and f (consisting of a homo-kinematical coupling 1-2 and
of a chain transmission 2-3) have the same kinematics as the previous
reducers; the efficiency relations and numerical values remain also
unchanged, excepting the numerical value of the size n0 which becomes
[[eta].sub.0] = 0.988. Therefore, the chain planetary reducers have the
following efficiencies' values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] < 0, the planetary reducers from Fig.
3,e and f don't need motors' brakes for the irreversible
transmission of power.

In conclusion, each of the variants SR5 and SR6 has the following
properties:

* the radial overall size is reduced in the case of the scheme f
and big in the case of the scheme e,

* it has a relative reduced axial overall size,

* the technology is relatively simple.

According to the first three properties, the requirements from the
list ([absolute value of i] = 100 [+ or -] 1.5 %, [eta] [greater than or
equal to] 0.4 and irreversible transmission of power) are fulfilled;
therefore, each of the variants SR5 and SR6 represents a solving
structure of the function FE2, too.

The principle solution or the motor-reducer concept will be
identified between the previous 6 solving structures (see Fig. 4).
Therefore, the generated structures are further ordered through a
technical and economical evaluation.

6. Evaluation of the solving structures and principle solution
establishment

The main features of the solving structures that were previously
considered are systematized in Fig. 4. Further, the remained solving
structures are ordered in Fig. 5, considering that the 4 criteria are of
different weights (fine evaluation). With this aim, first there were
established, in the Fig. 5,a the coefficients of absolute weight
([W.sub.k] = sum of the criterion's scores, see Fig. 5,a) and then
the coefficients of relative weight ([W.sub.k] =
[W.sub.k]/[SIGMA][W.sub.k], see Fig 5,a), knowing that: A [approximately
equal to] 4B [approximately equal to] 6C [approximately equal to] 8D.
With the marks from Fig. 5,b, it obviously results that the principle
solution of the motor-reducer is designated by the structure SR6 (whose
scheme is illustrated in Fig. 3,f).

7. Conclusions

From the exemplification of the generalized modeling algorithm
proposed by the authors (Diaconescu et al., 2008) in the case of the
conceptual design of a motor - reducer type product (which must drive a
rotational platform), the following conclusions can be drawn:

* In the first step, the motor-reducer global function is
identified and its subfunctions structure further developed.

* The next step deals with the generation of the solving variants
through the morphological matrix tool; in this example, 6 solving
structural variants are generated.

* The solving structures are next established based on their
geometrical and kinematical parameters synthesis and eliminating the
solving variants which don't fulfill the product requirements
quantitatively.

* In the last step, the remained solving structures are evaluated
using fine evaluation and hence the principle solution (SR6 structure)
of the motor-reducer is designated. This principle solution contains
three feasible modules (a motor without brake, a chain reducer and a
Schimdt pin semi-coupling) and represents the input entity in the
embodiment design phase.

DOI: 10.2507/daaam.scibook.2009.63

8. Acknowledgement

The authors will accomplish the design, manufacturing and testing
of the speed increaser for stand-alone hydropower stations in the
framework of the research project "Innovative mechatronic systems
for micro hydrostations, meant to the efficient exploitation of
hydrological potential from off-grid sites", ID_140. The
preparation and publishing of this paper would not have been possible
without the financial support of this research project.